Integrand size = 13, antiderivative size = 23 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 (c+d (a+b x))^{3/2}}{3 b d} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 (d (a+b x)+c)^{3/2}}{3 b d} \]
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Rule 32
Rule 33
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {c+d x} \, dx,x,a+b x\right )}{b} \\ & = \frac {2 (c+d (a+b x))^{3/2}}{3 b d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 (c+a d+b d x)^{3/2}}{3 b d} \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
gosper | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 d b}\) | \(20\) |
derivativedivides | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 d b}\) | \(20\) |
default | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 d b}\) | \(20\) |
trager | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 d b}\) | \(20\) |
pseudoelliptic | \(\frac {2 \left (c +d \left (b x +a \right )\right )^{\frac {3}{2}}}{3 b d}\) | \(20\) |
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none
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}}}{3 \, b d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (17) = 34\).
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \sqrt {c+d (a+b x)} \, dx=\begin {cases} \sqrt {c} x & \text {for}\: b = 0 \wedge d = 0 \\x \sqrt {a d + c} & \text {for}\: b = 0 \\\sqrt {c} x & \text {for}\: d = 0 \\\frac {2 a \sqrt {a d + b d x + c}}{3 b} + \frac {2 x \sqrt {a d + b d x + c}}{3} + \frac {2 c \sqrt {a d + b d x + c}}{3 b d} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}}}{3 \, b d} \]
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none
Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}}}{3 \, b d} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \sqrt {c+d (a+b x)} \, dx=\frac {2\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}}{3\,b\,d} \]
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